We are developing topological and lattice-based theoretical methods for studying the protein folding problem. This effort is unique inasmuch as: (i) we explore the full conformational space of the chain through exhaustive enumeration, and (ii) we explore the "general principles" of relationship between the native structure and amino acid sequence, through study of a large number of different sequences, and the full conformational space of each one. The studies are possible because we limit the conformational space: (i) by the short lengths of the lattice chains, and (ii) by constraints, either due to known neighbor/neighbor contacts, or by the high density in the globular state. For (ii), we are: (a) developing an "inference mechanics" to predict conformations from an incomplete set of known constraints, (b) simulating Hamiltonian walks, to predict all the conformations which fill a small region of space, and (c) applying path integral and other theoretical methods to the generalization of these results. Exhaustive simulation of open conformations of short lattice chains by Domb and Sykes 25 years ago have led to the modern theoretical developments in polymer physics, including scaling law theories, and path integral and renormalization group methods. We expect similar studies of the compact, rather than open, conformations to lead likewise to a "general principles" physics of compact molecules, principally proteins. Our preliminary results are exciting. (i) Using simulations and Feynman path integral methods, we have significantly improved on Jacobson-Stockmayer theory for stabilization of proteins due to cross-links like disulfides. (ii) A striking preliminary result is the finding that secondary structures in proteins arise from packing forces. Highly compact chains are unable to avoid formation of secondary structures. (iii) Some sequences have the potential to fold to native states (low-energy, high compactness, hydrophobic core), and others do not, depending partly on composition and partly on the "dispersal" of residues in the sequence. (iv) Proteins are predicted to have a high degree of "plasticity" in mutagenesis. Single-site changes are predicted to lead to minimal change in native structure and energy. Work of this type is of fundamental importance in biomedicine insofar as it is expected to have major impact or the protein folding problem, and on understanding structure/stability in proteins and other compact chain molecules.